Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{z^2 - 3z - 40}{z^2 - 5z - 24}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 - 3z - 40}{z^2 - 5z - 24} = \dfrac{(z + 5)(z - 8)}{(z + 3)(z - 8)} $ Notice that the term $(z - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z - 8)$ gives: $a = \dfrac{z + 5}{z + 3}$ Since we divided by $(z - 8)$, $z \neq 8$. $a = \dfrac{z + 5}{z + 3}; \space z \neq 8$